I don't see that this follows. It doesn't seem materially different than knowing that U always follows Q, and that J is always followed by a vowel in "legal" English language words.
I imagine it's technically possible to do this in a piecewise manner that doesn't "understand" the larger board. This could theoretically be done with number lines, and not a geometry (i.e. the 8x8 grid and current state of each square mentioned in the comment you replied to). It could also be done in a piecewise manner with three ternary numbers (e.g. 1,0,-1) for each 3 square sets.
I guess this is a kind of geometric representation on the order of Shannon's Theseus.
> It doesn't seem materially different than knowing that U always follows Q, and that J is always followed by a vowel in "legal" English language words.
The material difference is one of scale, not complexity.
Your rules have lookback = 1, while the Othello rules have lookback <= 63 and if you, say, are trying to play A1, you need to determine the current color of all squares on A1-A8, A1-H1, and A1-H8 (which is lookback <= 62) and then determine if one of 21 specific patterns exists.
Both can be technically be modeled with a lookup table, but for Othello that table would be size 3^63.
Then there becomes a "material" difference between Othello and those LL(1) grammars as grandparent comment suggested there wasn't.
I would argue the optimal compression for such a table is a representation of the geometric algorithm of determining move validity that all humans use intuitively, and speculate that any other compression algorithm below size say 1MB necessarily could be reduced to the geometric one.
In other words, Othello is a stateful, complex game, so if GPT is doing validation efficiently, it necessarily encoded something that unequivocally can be described as the "geometric structure".
https://content.wolfram.com/uploads/sites/43/2023/02/sw02142... from https://writings.stephenwolfram.com/2023/02/what-is-chatgpt-...
I imagine it's technically possible to do this in a piecewise manner that doesn't "understand" the larger board. This could theoretically be done with number lines, and not a geometry (i.e. the 8x8 grid and current state of each square mentioned in the comment you replied to). It could also be done in a piecewise manner with three ternary numbers (e.g. 1,0,-1) for each 3 square sets.
I guess this is a kind of geometric representation on the order of Shannon's Theseus.